Calculators
Multinomial Coefficient Calculator
Calculate exact multinomial coefficients and distinct arrangements of repeated items from group counts, privately in your browser.
Count arrangements when items within each group are identical. Enter up to 100 group counts separated by commas or whitespace, with a total of at most 10000 items. Calculations stay in your browser.
Count arrangements of repeated items.
For group counts n₁, n₂, … with total N, the multinomial coefficient is N! / (n₁! × n₂! × …). It counts arrangements of N items when items of the same type are identical and different types are distinguishable. Counts 2, 2, 1 give 5! / (2! × 2! × 1!) = 30 arrangements, such as the distinct rearrangements of AABBC. It also counts assignments of N distinct items to labeled groups of those fixed sizes. Zero-sized groups contribute 0! = 1; all zero counts give one empty arrangement. One group gives one arrangement. Enter 1–100 nonnegative integer counts, each at most 10000, with a total at most 10000. Separate counts by commas or whitespace; signs, decimals and scientific notation are rejected. Leading zeros and surrounding whitespace are accepted. Exact BigInt arithmetic evaluates a product of binomial coefficients without rounding or factorial intermediates. This is a count, not a multinomial probability: probabilities additionally require a probability for each type. Inputs stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a multinomial coefficient?
For counts totaling N, it is N! divided by the product of the factorials of each count. It counts arrangements with identical items within each type.
How many arrangements does AABBC have?
The group counts are 2, 2, 1, so the count is 5! / (2! × 2! × 1!) = 30.
Are zero counts allowed?
Yes. Zero factorial is one, so zero-sized groups do not change the count. If all counts are zero, there is one empty arrangement.
Does this calculate multinomial probabilities?
No. This gives the exact coefficient. A multinomial probability also multiplies by the probability of each type raised to its count.
What are the limits and are results private?
Enter up to 100 group counts with a total no greater than 10000. Browser BigInt arithmetic retains every digit and no input is uploaded.